John E. Post, KA5GSQ Embry-Riddle Aeronautical University, 3700 Willow Creek Rd, Prescott, AZ, 86301; [email protected] On Determining Loop Gain through Circuit Simulation Loop gain is a fundamental parameter for electronic circuits that employ either positive or negative feedback. This article discusses how to determine loop gain through circuit simulation. Amplifiers employing positive or nega- tive feedback are fundamental building blocks in electronic circuits. Negative feed- back is employed to linearize amplifiers in order to reduce distortion of the input signal and improve amplifier bandwidth. Conversely, applying sufficient positive feedback to an amplifier results in an oscilla- tor because an output signal occurs when no Figure 1 — Block diagram of positive 1 feedback amplifier depicting input voltage input signal is present. vi, output voltage vo, and feedback voltage Figure 1 depicts a block diagram for a v . simple amplifier with positive feedback. The f Figure 3 — Alternative model of a gain of the amplifier is A while the gain (or positive feedback amplifier consisting of loss) of the feedback network is K. The input a dependent current source with current and output voltages of the overall circuit are gmvx in parallel with a feedback impedance v and v respectively, while the voltage at the Zf. Each “X” indicates locations where it is i o possible to break open the loop in order output of the feedback network is vf. Again to determine loop gain. referring to Figure 1, the output voltage is obtained from the amplified sum of the input and feedback voltages, or electronic circuit will actually function as an oscillator. v= A v + v Eq. 1 o( i f ) Figure 2 — Positive feedback amplifier In order to develop a method to deter- redrawn to emphasize the feedback loop. mine the loop gain, consider Figure 2, which while the voltage at the output of the feed- Each “X” indicates locations where it is is redrawn to emphasize the feedback por- back network is found from possible to break open the loop in order tion of the loop from Figure 1. If the loop is vf = Kvo Eq. 2 to determine loop gain. opened at either “X” and a test voltage vtest Combining Equations 1 and 2 in order to is applied to the side of the loop that leads eliminate the feedback voltage term allows fier to sustain oscillations the Barkhausen to the amplifier input, what voltage vmeasure is the voltage at the output of the amplifier to criteria requires that the magnitude of T measured at the other side of the loop open- be expressed in terms of the input voltage as is equal to unity at a frequency where the ing? Obviously vmeasure = AKvtest so that the A 1 1 phase of T is equal to 0 degrees or radi- loop voltage gain is v /v = AK = T. v =v = v measure test o i 1 i ans (or some integer multiple of 360o Figure 3 shows a circuit model of the pos- 1− A K K − 1 Eq. 3 AK or 2π radians). In order for oscillations to itive feedback amplifier, Figure 2. As shown In Equation 3 the term AK is referred to start the loop gain must be greater than unity in the figure the circuit consists of a depen- as the loop gain T. at a frequency where the phase is 0 degrees dent source, in this case a voltage controlled o In order for the positive feedback ampli- or radians (or some integer multiple of 360 current source that represents an ideal trans- 2 or 2π radians). Thus, the loop gain provides conductance amplifier and an impedance Zf critical information concerning whether an 1Notes appear on page 7. that represents the load of the feedback net- QEX – March/April 2011 3 work on the controlled source. In general it is in the left side of Figure 5. This time an arbi- overall loop gain T from the loop voltage possible to define either a loop voltage gain trary current source iz is added in parallel gain Tv and loop current gain Ti. This is Tv or a loop current gain Ti. The procedure with the dependent source and feedback net- demonstrated in Appendix 1, which modi- for determining either one is the same—the work. Because the internal impedance of the fies Middlebrook’s result for the case of a loop is opened, a test voltage (current) is current source is infinite the load impedance positive feedback amplifier (oscillator). As applied and a response voltage (current) is seen by the dependent source is unchanged shown in Appendix 1, once the loop volt- determined and then their ratio is obtained. and the bias currents are unaffected. In this age gain Tv and the loop current gain Ti are But now a second load impedance Zf is con- case it is necessary to obtain the currents ix obtained the overall loop gain T can be found nected across the source in order to maintain and iy, through either measurement or circuit from the same load impedance on the source simulation, and then determine the loop cur- Tv T i − 1 with the loop open as with the loop closed. T = rent gain Ti = iy/ix. T+ T − 2 This is shown in the two circuits shown in Next it is necessary to determine the v i Eq. A-1 Figure 4. For the circuit on the left side of the figure applying a test voltage of vx results in a voltage of vy and the loop voltage gain is Tv = vy/vx = gmvxZf /vx =gmZf. For the circuit on the right side of the figure applying a test current of ix results in a current of iy and the loop current gain is Ti = iy/ix = gm(vx/Zf ) = gmZf . In this simple example Tv = Ti = gmZf = T which is the overall loop gain, but this result holds only if the output impedance of the dependent source is ignored as we shall see. Although the technique of opening the loop appears simple in theory there are a number of factors that complicate its appli- Figure 4 — In order to obtain loop voltage gain break loop at “X”, insert a test voltage cation to a practical circuit. For example source, and then determine Tv = vy/vx (left circuit). In order to obtain loop current gain break loop at “X”, insert a test current source, and then determine Ti = iy/ix (right circuit). opening the loop may remove the dc bias In each case after the loop is broken it is necessary to insert a replacement feedback current for the active device so that it no lon- impedance Zf in order to maintain a constant load for the dependent current source. ger functions. Another problem is precisely duplicating the value of the load impedance that appears in parallel with the dependent source once the loop is broken. An Experimental Method for Determining the Loop Gain In 1975 R.D. Middlebrook proposed a method to overcome these limitations for negative feedback amplifiers.3 Middlebrook applied this method to measure the loop gain of negative feedback amplifiers. It is also possible to apply the method to determine the loop gain of negative feedback amplifiers 4 through circuit simulation. Figure 5 — Determine loop voltage gain by inserting an arbitrary voltage source vz and In general Middlebrook’s method deter- then obtaining Tv = vy/vx (right circuit). Determining loop current gain by inserting an arbitrary current source iz and then obtaining Ti = iy/ix (left circuit). mines the loop voltage gain Tv by inserting an arbitrary voltage source vz into the loop as shown by the circuit on the right side of Figure 5. Because practical dependent sources contain an internal impedance Zs this impedance, as well as the load impedance Zf due to the feedback network, is shown in the circuits in Figure 5. Because the internal impedance of the voltage source vz is a short circuit the load impedance seen by the dependent source is unchanged and bias currents are unaffected. All that remains is to obtain the voltages vx and vy, through either measurement or circuit simulation, and then determine the loop volt- age gain Tv =vy/vx. The loop current gain Ti may be obtained Figure 6 — Circuit simulation to determine loop voltage and current gain for the circuits by the dual of the previous method as shown shown in Figure 5 for the case of Zs = Rs = 18 Ω and Zf = Rf = 18 Ω. 4 QEX – March/April 2011 For large values of Tv and Ti Equation A-1 can be approximated as TT T @ v i TTv+ i which demonstrates that, like the formula for two parallel resistors, the overall loop gain T will be smaller than the smallest of either the loop current gain Ti or the loop voltage gain Tv. The balance of this article will demon- strate how to apply Equation A-1 through cir- cuit simulation in the case of dc, audio, and radio frequency positive feedback amplifiers. Figure 7 — Circuit simulation to determine loop voltage and current gain for the circuits shown in Figure 5 for the case of Zs = Rs = 10 Ω and Zf = Rf = 90 Ω. Example: Determining the Loop Gain for DC Circuits Because Equation A-1 was derived by applying general circuit laws, it applies equally to either time varying (ac) or con- stant (dc) voltages. Thus before applying the method to determine the loop gain of an oscillator it is instructive to review sev- eral dc simulations to demonstrate that Equation A-1 is correct, as well as illustrate the method of obtaining the loop gain T through simulation.